Hi, Jeff —
The bells are rung when the Words of Institution are said — these are the words that Jesus said
to consecrate the first Eucharist. We believe, then,
that when the priest says:
This is My Body and This is the [Cup|Chalice] of My Blood,
the bread is transformed into the Body of Christ,
and the blood is transformed into the Blood
of Christ. (To
be technical, the Body and Blood, Soul and
Divinity of Christ are contained in each form: the Consecrated Bread and Consecrated Wine.) Therefore it is a pious custom
to ring the bells at those times.
I believe the bells are also rung at the calling
down of the Holy Spirit, or Epiclesis.
In the older
liturgy, these prayers were done silently, so the
bells told the faithful when the transformation had
taken place; now they merely accentuate it.